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<h1 id="Weak-Galerkin-Methods-for-Poisson-Equation-in-2D">Weak Galerkin Methods for Poisson Equation in 2D<a class="anchor-link" href="#Weak-Galerkin-Methods-for-Poisson-Equation-in-2D">&#182;</a></h1>
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<p>This example is to show the rate of convergence of the lowest order Weak Galerkin finite element
approximation of the Poisson equation on the unit square:</p>
$$- \Delta u = f \; \hbox{in } (0,1)^2$$<p>for the following boundary conditions</p>
<ul>
<li>Non-empty Dirichlet boundary condition: $u=g_D \hbox{ on }\Gamma_D, \nabla u\cdot n=g_N \hbox{ on }\Gamma_N.$</li>
<li>Pure Neumann boundary condition: $\nabla u\cdot n=g_N \hbox{ on } \partial \Omega$.</li>
<li>Robin boundary condition: $g_R u + \nabla u\cdot n=g_N \hbox{ on }\partial \Omega$.</li>
</ul>
<p><strong>References</strong>:</p>
<ul>
<li><a href="femdoc.html">Quick Introduction to Finite Element Methods</a></li>
<li><a href="http://www.math.uci.edu/~chenlong/226/Ch2FEM.pdf">Introduction to Finite Element Methods</a></li>
<li><a href="http://www.math.uci.edu/~chenlong/ifemdoc/fem/WGprogramming.pdf">Progamming of Weak Galerkin Methods</a></li>
</ul>
<p><strong>Subroutines</strong>:</p>

<pre><code>- PoissonWG
- squarePoissonWG
- femPoisson
- PoissonWGfemrate

</code></pre>
<p>The method is implemented in <code>PoissonWG</code> subroutine and can be tested in <code>squarePoissonWG</code>. Together with other elements (P1, P2, P3, Q1), <code>femPoisson</code> provides a concise interface to solve Poisson equation. The P2 element is tested in <code>PoissonWGfemrate</code>. This doc is based on <code>PoissonWGfemrate</code>.</p>

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<h2 id="The-Lowest-Order-Weak-Galerkin-Element">The Lowest Order Weak Galerkin Element<a class="anchor-link" href="#The-Lowest-Order-Weak-Galerkin-Element">&#182;</a></h2><p>The basis and the local matrices can be found in <a href="http://www.math.uci.edu/~chenlong/ifemdoc/fem/WGprogramming.pdf">Progamming of Weak Galerkin Methods</a></p>

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<h2 id="Mixed-boundary-condition">Mixed boundary condition<a class="anchor-link" href="#Mixed-boundary-condition">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Setting</span>
<span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">]</span> <span class="p">=</span> <span class="n">squaremesh</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span><span class="mf">0.25</span><span class="p">);</span> 
<span class="n">mesh</span> <span class="p">=</span> <span class="n">struct</span><span class="p">(</span><span class="s">&#39;node&#39;</span><span class="p">,</span><span class="n">node</span><span class="p">,</span><span class="s">&#39;elem&#39;</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">option</span><span class="p">.</span><span class="n">L0</span> <span class="p">=</span> <span class="mi">2</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">maxIt</span> <span class="p">=</span> <span class="mi">4</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">printlevel</span> <span class="p">=</span> <span class="mi">1</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">elemType</span> <span class="p">=</span> <span class="s">&#39;WG&#39;</span><span class="p">;</span>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">% Mixed boundary condition</span>
<span class="n">pde</span> <span class="p">=</span> <span class="n">sincosdata</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Dirichlet&#39;</span><span class="p">,</span><span class="s">&#39;~(x==0)&#39;</span><span class="p">,</span><span class="s">&#39;Neumann&#39;</span><span class="p">,</span><span class="s">&#39;x==0&#39;</span><span class="p">);</span>
<span class="n">femPoisson</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     3136,  #nnz:    14866, smoothing: (1,1), iter: 12,   err = 2.67e-09,   time =  0.1 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    12416,  #nnz:    55426, smoothing: (1,1), iter: 12,   err = 2.62e-09,   time = 0.097 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    49408,  #nnz:   238770, smoothing: (1,1), iter: 12,   err = 2.57e-09,   time = 0.18 s

 #Dof       h        ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

  800   6.250e-02   1.71814e-03   1.62480e-01   6.33304e-02   3.20790e-03
 3136   3.125e-02   4.29635e-04   8.12679e-02   3.15253e-02   8.03198e-04
12416   1.562e-02   1.07415e-04   4.06374e-02   1.57451e-02   2.00838e-04
49408   7.812e-03   2.68539e-05   2.03191e-02   7.87036e-03   5.02182e-05

 #Dof   Assemble     Solve      Error      Mesh    

  800   9.00e-02   1.41e-02   9.00e-02   1.00e-02
 3136   7.00e-02   1.03e-01   4.00e-02   2.00e-02
12416   1.30e-01   9.69e-02   5.00e-02   1.00e-01
49408   3.00e-01   1.83e-01   1.10e-01   1.00e-01


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<h2 id="Pure-Neumann-boundary-condition">Pure Neumann boundary condition<a class="anchor-link" href="#Pure-Neumann-boundary-condition">&#182;</a></h2><p>When pure Neumann boundary condition is posed, i.e., $-\Delta u =f$ in $\Omega$ and $\nabla u\cdot n=g_N$ on $\partial \Omega$, the data should be consisitent in the sense that $\int_{\Omega} f \, dx + \int_{\partial \Omega} g \, ds = 0$. The solution is unique up to a constant. A post-process is applied such that the constraint $\int_{\Omega}u_h dx = 0$ is imposed.</p>

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<div class=" highlight hl-matlab"><pre><span></span><span class="n">option</span><span class="p">.</span><span class="n">plotflag</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="n">pde</span> <span class="p">=</span> <span class="n">sincosNeumanndata</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Neumann&#39;</span><span class="p">);</span>
<span class="n">femPoisson</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     3136,  #nnz:    15339, smoothing: (1,1), iter: 13,   err = 4.49e-09,   time = 0.066 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    12416,  #nnz:    56235, smoothing: (1,1), iter: 14,   err = 1.90e-09,   time = 0.07 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    49408,  #nnz:   240683, smoothing: (1,1), iter: 14,   err = 3.96e-09,   time = 0.24 s

 #Dof       h        ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

  800   6.250e-02   1.76689e-02   6.83050e-01   5.49515e-01   2.27492e-01
 3136   3.125e-02   4.83700e-03   4.13444e-01   2.27292e-01   1.18952e-01
12416   1.562e-02   1.28283e-03   2.09935e-01   1.07773e-01   7.44907e-02
49408   7.812e-03   3.30853e-04   1.05872e-01   5.43084e-02   4.20973e-02

 #Dof   Assemble     Solve      Error      Mesh    

  800   7.00e-02   1.42e-03   3.00e-02   0.00e+00
 3136   1.00e-02   6.63e-02   2.00e-02   0.00e+00
12416   7.00e-02   7.00e-02   3.00e-02   1.00e-02
49408   2.70e-01   2.41e-01   1.30e-01   6.00e-02


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<h2 id="Robin-boundary-condition">Robin boundary condition<a class="anchor-link" href="#Robin-boundary-condition">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">option</span><span class="p">.</span><span class="n">plotflag</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="n">pde</span> <span class="p">=</span> <span class="n">sincosRobindata</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Robin&#39;</span><span class="p">);</span>
<span class="n">femPoisson</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     3136,  #nnz:    15344, smoothing: (1,1), iter: 12,   err = 1.81e-09,   time = 0.05 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    12416,  #nnz:    56240, smoothing: (1,1), iter: 11,   err = 9.88e-09,   time = 0.051 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    49408,  #nnz:   240688, smoothing: (1,1), iter: 12,   err = 1.78e-09,   time = 0.18 s

 #Dof       h        ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

  800   6.250e-02   8.89017e-03   6.48648e-01   2.60943e-01   1.69325e-02
 3136   3.125e-02   2.23100e-03   3.24914e-01   1.27079e-01   4.31205e-03
12416   1.562e-02   5.58282e-04   1.62530e-01   6.31035e-02   1.08242e-03
49408   7.812e-03   1.39604e-04   8.12741e-02   3.14968e-02   2.70809e-04

 #Dof   Assemble     Solve      Error      Mesh    

  800   4.00e-02   1.37e-03   0.00e+00   1.00e-02
 3136   2.00e-02   5.02e-02   1.00e-02   1.00e-02
12416   5.00e-02   5.15e-02   5.00e-02   1.00e-02
49408   2.10e-01   1.81e-01   1.00e-01   6.00e-02


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"
>
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<h2 id="Conclusion">Conclusion<a class="anchor-link" href="#Conclusion">&#182;</a></h2><p>The optimal rate of convergence of the H1-norm (1st order) and L2-norm
(2nd order) is observed. No superconvergence for $\|\nabla u_I - \nabla u_h\|$.</p>
<p>MGCG converges uniformly in all cases.</p>

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